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Adapt to https://github.com/coq/coq/pull/18590 #42

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Jan 31, 2024
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Adapt to https://github.com/coq/coq/pull/18590 #42

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18 changes: 9 additions & 9 deletions theories/Classes.v
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Expand Up @@ -25,7 +25,7 @@ Class Graph := {
T: Type;
X: T -> T -> Type;
equal: forall A B, relation (X A B);
equal_:> forall A B, Equivalence (equal A B)
equal_:: forall A B, Equivalence (equal A B)
}.

(*Arguments equal : simpl never.*)
Expand Down Expand Up @@ -91,31 +91,31 @@ Section Structures.
Context {Mo: Monoid_Ops G} {SLo: SemiLattice_Ops G} {Ko: Star_Op G} {Co: Converse_Op G}.

Class Monoid := {
dot_compat:> forall A B C, Proper (equal A B ==> equal B C ==> equal A C) (dot A B C);
dot_compat:: forall A B C, Proper (equal A B ==> equal B C ==> equal A C) (dot A B C);
dot_assoc: forall A B C D (x: X A B) y (z: X C D), x*(y*z) == (x*y)*z;
dot_neutral_left: forall A B (x: X A B), 1*x == x;
dot_neutral_right: forall A B (x: X B A), x*1 == x
}.

Class SemiLattice := {
plus_compat:> forall A B, Proper (equal A B ==> equal A B ==> equal A B) (plus A B);
plus_compat:: forall A B, Proper (equal A B ==> equal A B ==> equal A B) (plus A B);
plus_neutral_left: forall A B (x: X A B), 0+x == x;
plus_idem: forall A B (x: X A B), x+x == x;
plus_assoc: forall A B (x y z: X A B), x+(y+z) == (x+y)+z;
plus_com: forall A B (x y: X A B), x+y == y+x
}.

Class IdemSemiRing := {
ISR_Monoid :> Monoid;
ISR_SemiLattice :> SemiLattice;
ISR_Monoid :: Monoid;
ISR_SemiLattice :: SemiLattice;
dot_ann_left: forall A B C (x: X B C), zero A B * x == 0;
dot_ann_right: forall A B C (x: X C B), x * zero B A == 0;
dot_distr_left: forall A B C (x y: X A B) (z: X B C), (x+y)*z == x*z + y*z;
dot_distr_right: forall A B C (x y: X B A) (z: X C B), z*(x+y) == z*x + z*y
}.

Class KleeneAlgebra := {
KA_ISR :> IdemSemiRing;
KA_ISR :: IdemSemiRing;
star_make_left: forall A (a:X A A), 1 + a#*a == a#;
star_destruct_left: forall A B (a: X A A) (c: X A B), a*c <== c -> a#*c <== c;
star_destruct_right: forall A B (a: X A A) (c: X B A), c*a <== c -> c*a# <== c
Expand All @@ -127,12 +127,12 @@ Section Structures.
(* TODO: introduce an intermediate ConverseMonoid class *)

Class ConverseIdemSemiRing := {
CISR_SL :> SemiLattice;
CISR_SL :: SemiLattice;
dot_compat_c: forall A B C, Proper (equal A B ==> equal B C ==> equal A C) (dot A B C);
dot_assoc_c: forall A B C D (x: X A B) y (z: X C D), x*(y*z) == (x*y)*z;
dot_neutral_left_c: forall A B (x: X A B), 1*x == x;

conv_compat:> forall A B, Proper (equal A B ==> equal B A) (conv A B);
conv_compat:: forall A B, Proper (equal A B ==> equal B A) (conv A B);
conv_invol: forall A B (x: X A B), x`` == x;
conv_dot: forall A B C (x: X A B) (y: X B C), (x*y)` == y`*x`;
conv_plus: forall A B (x y: X A B), (x+y)` == y`+x`;
Expand All @@ -141,7 +141,7 @@ Section Structures.
}.

Class ConverseKleeneAlgebra := {
CKA_CISR :> ConverseIdemSemiRing;
CKA_CISR :: ConverseIdemSemiRing;
star_make_left_c: forall A (a:X A A), 1 + a#*a == a#;
star_destruct_left_c: forall A B (a: X A A) (c: X A B), a*c <== c -> a#*c <== c
}.
Expand Down
2 changes: 1 addition & 1 deletion theories/DKA_DFA_Equiv.v
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Expand Up @@ -182,7 +182,7 @@ Section correctness.

Class invariant tarjan : Prop :=
{
i_wf_tarjan :> DS.WF tarjan ;
i_wf_tarjan :: DS.WF tarjan ;
i_final : forall x y, {{tarjan}} x y -> final x = final y
}.

Expand Down
10 changes: 5 additions & 5 deletions theories/Functors.v
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Expand Up @@ -36,13 +36,13 @@ Section Defs.
forall A B y, exists x, F A B x == y.

Class monoid_functor {Mo1: Monoid_Ops G1} {Mo2: Monoid_Ops G2} (F: functor G1 G2) := {
monoid_graph_functor :> graph_functor F;
monoid_graph_functor :: graph_functor F;
functor_dot : forall A B C x y, F A C (x*y) == F A B x * F B C y;
functor_one : forall A, F A A 1 == 1
}.

Class semilattice_functor {SLo1: SemiLattice_Ops G1} {SL2: SemiLattice_Ops G2} (F: functor G1 G2) := {
semilattice_graph_functor :> graph_functor F;
semilattice_graph_functor :: graph_functor F;
functor_plus : forall A B x y, F A B (x+y) == F A B x + F A B y;
functor_zero : forall A B, F A B 0 == 0
}.
Expand Down Expand Up @@ -71,8 +71,8 @@ Section Defs.
{Ko1: Star_Op G1} {Ko2: Star_Op G2}.

Class semiring_functor (F: functor G1 G2) := {
semiring_monoid_functor :> monoid_functor F;
semiring_semilattice_functor :> semilattice_functor F
semiring_monoid_functor :: monoid_functor F;
semiring_semilattice_functor :: semilattice_functor F
}.

Lemma functor_star_leq {KA1: KleeneAlgebra G1} {KA2: KleeneAlgebra G2}
Expand All @@ -88,7 +88,7 @@ Section Defs.
Qed.

Class kleene_functor (F: functor G1 G2) := {
kleene_semiring :> semiring_functor F;
kleene_semiring :: semiring_functor F;
functor_star: forall A a, F A A (a#) == (F A A a) #
}.

Expand Down
4 changes: 2 additions & 2 deletions theories/StrictKleeneAlgebra.v
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Expand Up @@ -41,9 +41,9 @@ Delimit Scope SA_scope with SA.

(** Strict Kleene Algebras axioms *)
Class StrictKleeneAlgebra {G: Graph} {Ops: SKA_Ops G} := {
dot_compat:>
dot_compat::
forall A B C, Proper (equal A B ==> equal B C ==> equal A C) (dot A B C);
plus_compat:>
plus_compat::
forall A B, Proper (equal A B ==> equal A B ==> equal A B) (plus A B);
dot_assoc: forall A B C D (x: X A B) y (z: X C D), x*(y*z) == (x*y)*z;
dot_neutral_left: forall A B (x: X A B), 1*x == x;
Expand Down
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